Why are these lines tangent?

Question

I was trying the problems at http://euclidthegame.org and for level 20, ending up using, but couldn't see the reason behind the following:

We have a circle centred on B and a point A outside the circle. Construct E at the midpoint of AB. Draw a circle centred on E passing through A and B. F and G are the intersection points of the two circles. Now the lines AF and AG are tangents to the original circle. How can we see that is the case?

Answer 1

Since $AB$ is the diameter of the 2nd circle, $\angle AGB=\frac{\pi}{2}$. This means that $AG \perp BG$ which is the radius of the 1st circle.

Answer 2

It suffices to show that the line $(BG)\perp (AG)$. Notice that the triangles $AEG$ and $EBG$ are isosceles so

$$\angle{EAG}=\angle EGA$$ and $$\angle EGB=\angle EBG$$ so we have

$$\angle AGB=\angle AGE+\angle EGB=\frac12\angle AEB=\frac{180^°}{2}=90^°$$ hence the line $(AG)$ is tangent to the circle on $G$.